pathfinder/content/src/outline.rs

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// pathfinder/content/src/outline.rs
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//
// Copyright © 2019 The Pathfinder Project Developers.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
//! A compressed in-memory representation of paths.
use crate::clip::{self, ContourPolygonClipper, ContourRectClipper};
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use crate::dilation::ContourDilator;
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use crate::orientation::Orientation;
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use crate::segment::{Segment, SegmentFlags, SegmentKind};
use pathfinder_geometry::line_segment::LineSegment2F;
use pathfinder_geometry::rect::RectF;
use pathfinder_geometry::transform2d::Transform2F;
use pathfinder_geometry::transform3d::Perspective;
use pathfinder_geometry::unit_vector::UnitVector;
use pathfinder_geometry::vector::{Vector2F, vec2f};
use std::f32::consts::PI;
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use std::fmt::{self, Debug, Formatter};
use std::mem;
#[derive(Clone)]
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pub struct Outline {
pub(crate) contours: Vec<Contour>,
pub(crate) bounds: RectF,
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}
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#[derive(Clone)]
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pub struct Contour {
pub(crate) points: Vec<Vector2F>,
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pub(crate) flags: Vec<PointFlags>,
pub(crate) bounds: RectF,
pub(crate) closed: bool,
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}
bitflags! {
pub struct PointFlags: u8 {
const CONTROL_POINT_0 = 0x01;
const CONTROL_POINT_1 = 0x02;
}
}
bitflags! {
pub struct PushSegmentFlags: u8 {
const UPDATE_BOUNDS = 0x01;
const INCLUDE_FROM_POINT = 0x02;
}
}
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impl Outline {
#[inline]
pub fn new() -> Outline {
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Outline {
contours: vec![],
bounds: RectF::default(),
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}
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}
#[inline]
pub fn from_segments<I>(segments: I) -> Outline
where
I: Iterator<Item = Segment>,
{
let mut outline = Outline::new();
let mut current_contour = Contour::new();
for segment in segments {
if segment.flags.contains(SegmentFlags::FIRST_IN_SUBPATH) {
if !current_contour.is_empty() {
outline
.contours
.push(mem::replace(&mut current_contour, Contour::new()));
}
current_contour.push_point(segment.baseline.from(), PointFlags::empty(), true);
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}
if segment.flags.contains(SegmentFlags::CLOSES_SUBPATH) {
if !current_contour.is_empty() {
current_contour.close();
let contour = mem::replace(&mut current_contour, Contour::new());
outline.push_contour(contour);
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}
continue;
}
if segment.is_none() {
continue;
}
if !segment.is_line() {
current_contour.push_point(segment.ctrl.from(), PointFlags::CONTROL_POINT_0, true);
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if !segment.is_quadratic() {
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current_contour.push_point(
segment.ctrl.to(),
PointFlags::CONTROL_POINT_1,
true,
);
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}
}
current_contour.push_point(segment.baseline.to(), PointFlags::empty(), true);
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}
outline.push_contour(current_contour);
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outline
}
#[inline]
pub fn bounds(&self) -> RectF {
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self.bounds
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}
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#[inline]
pub fn contours(&self) -> &[Contour] {
&self.contours
}
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#[inline]
pub fn into_contours(self) -> Vec<Contour> {
self.contours
}
/// Removes all contours from this outline.
#[inline]
pub fn clear(&mut self) {
self.contours.clear();
self.bounds = RectF::default();
}
pub fn push_contour(&mut self, contour: Contour) {
if contour.is_empty() {
return;
}
if self.contours.is_empty() {
self.bounds = contour.bounds;
} else {
self.bounds = self.bounds.union_rect(contour.bounds);
}
self.contours.push(contour);
}
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pub fn pop_contour(&mut self) -> Option<Contour> {
let last_contour = self.contours.pop();
let mut new_bounds = None;
for contour in &mut self.contours {
contour.update_bounds(&mut new_bounds);
}
self.bounds = new_bounds.unwrap_or_else(|| RectF::default());
last_contour
}
pub fn transform(&mut self, transform: &Transform2F) {
if transform.is_identity() {
return;
}
let mut new_bounds = None;
for contour in &mut self.contours {
contour.transform(transform);
contour.update_bounds(&mut new_bounds);
}
self.bounds = new_bounds.unwrap_or_else(|| RectF::default());
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}
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pub fn apply_perspective(&mut self, perspective: &Perspective) {
let mut new_bounds = None;
for contour in &mut self.contours {
contour.apply_perspective(perspective);
contour.update_bounds(&mut new_bounds);
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}
self.bounds = new_bounds.unwrap_or_else(|| RectF::default());
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}
pub fn dilate(&mut self, amount: Vector2F) {
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let orientation = Orientation::from_outline(self);
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self.contours
.iter_mut()
.for_each(|contour| contour.dilate(amount, orientation));
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self.bounds = self.bounds.dilate(amount);
}
pub fn prepare_for_tiling(&mut self, view_box: RectF) {
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self.contours
.iter_mut()
.for_each(|contour| contour.prepare_for_tiling(view_box));
self.bounds = self
.bounds
.intersection(view_box)
.unwrap_or_else(|| RectF::default());
}
pub fn is_outside_polygon(&self, clip_polygon: &[Vector2F]) -> bool {
clip::rect_is_outside_polygon(self.bounds, clip_polygon)
}
fn is_inside_polygon(&self, clip_polygon: &[Vector2F]) -> bool {
clip::rect_is_inside_polygon(self.bounds, clip_polygon)
}
pub fn clip_against_polygon(&mut self, clip_polygon: &[Vector2F]) {
// Quick check.
if self.is_inside_polygon(clip_polygon) {
return;
}
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for contour in mem::replace(&mut self.contours, vec![]) {
self.push_contour(ContourPolygonClipper::new(clip_polygon, contour).clip());
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}
}
pub fn clip_against_rect(&mut self, clip_rect: RectF) {
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if clip_rect.contains_rect(self.bounds) {
return;
}
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for contour in mem::replace(&mut self.contours, vec![]) {
self.push_contour(ContourRectClipper::new(clip_rect, contour).clip());
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}
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}
#[inline]
pub fn close_all_contours(&mut self) {
self.contours.iter_mut().for_each(|contour| contour.close());
}
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}
impl Debug for Outline {
fn fmt(&self, formatter: &mut Formatter) -> fmt::Result {
for (contour_index, contour) in self.contours.iter().enumerate() {
if contour_index > 0 {
write!(formatter, " ")?;
}
contour.fmt(formatter)?;
}
Ok(())
}
}
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impl Contour {
#[inline]
pub fn new() -> Contour {
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Contour {
points: vec![],
flags: vec![],
bounds: RectF::default(),
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closed: false,
}
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}
#[inline]
pub fn with_capacity(length: usize) -> Contour {
Contour {
points: Vec::with_capacity(length),
flags: Vec::with_capacity(length),
bounds: RectF::default(),
closed: false,
}
}
// Replaces this contour with a new one, with arrays preallocated to match `self`.
#[inline]
pub(crate) fn take(&mut self) -> Contour {
let length = self.len() as usize;
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mem::replace(
self,
Contour {
points: Vec::with_capacity(length),
flags: Vec::with_capacity(length),
bounds: RectF::default(),
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closed: false,
},
)
}
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/// restore self to the state of Contour::new(), but keep the points buffer allocated
#[inline]
pub fn clear(&mut self) {
self.points.clear();
self.flags.clear();
self.bounds = RectF::default();
self.closed = false;
}
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#[inline]
pub fn iter(&self, flags: ContourIterFlags) -> ContourIter {
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ContourIter {
contour: self,
index: 1,
flags,
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}
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}
#[inline]
pub fn is_empty(&self) -> bool {
self.points.is_empty()
}
#[inline]
pub fn len(&self) -> u32 {
self.points.len() as u32
}
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#[inline]
pub fn bounds(&self) -> RectF {
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self.bounds
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}
#[inline]
pub fn is_closed(&self) -> bool {
self.closed
}
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#[inline]
pub fn position_of(&self, index: u32) -> Vector2F {
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self.points[index as usize]
}
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#[inline]
pub fn last_position(&self) -> Option<Vector2F> {
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self.points.last().cloned()
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}
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#[inline]
pub(crate) fn position_of_last(&self, index: u32) -> Vector2F {
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self.points[self.points.len() - index as usize]
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}
#[inline]
pub fn push_endpoint(&mut self, point: Vector2F) {
self.push_point(point, PointFlags::empty(), true);
}
#[inline]
pub fn push_quadratic(&mut self, ctrl: Vector2F, point: Vector2F) {
self.push_point(ctrl, PointFlags::CONTROL_POINT_0, true);
self.push_point(point, PointFlags::empty(), true);
}
#[inline]
pub fn push_cubic(&mut self, ctrl0: Vector2F, ctrl1: Vector2F, point: Vector2F) {
self.push_point(ctrl0, PointFlags::CONTROL_POINT_0, true);
self.push_point(ctrl1, PointFlags::CONTROL_POINT_1, true);
self.push_point(point, PointFlags::empty(), true);
}
#[inline]
pub fn close(&mut self) {
self.closed = true;
}
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#[inline]
pub(crate) fn push_point(&mut self,
point: Vector2F,
flags: PointFlags,
update_bounds: bool) {
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debug_assert!(!point.x().is_nan() && !point.y().is_nan());
if update_bounds {
let first = self.is_empty();
union_rect(&mut self.bounds, point, first);
}
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self.points.push(point);
self.flags.push(flags);
}
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#[inline]
pub(crate) fn push_segment(&mut self, segment: &Segment, flags: PushSegmentFlags) {
if segment.is_none() {
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return;
}
let update_bounds = flags.contains(PushSegmentFlags::UPDATE_BOUNDS);
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self.push_point(segment.baseline.from(), PointFlags::empty(), update_bounds);
if !segment.is_line() {
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self.push_point(
segment.ctrl.from(),
PointFlags::CONTROL_POINT_0,
update_bounds,
);
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if !segment.is_quadratic() {
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self.push_point(
segment.ctrl.to(),
PointFlags::CONTROL_POINT_1,
update_bounds,
);
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}
}
self.push_point(segment.baseline.to(), PointFlags::empty(), update_bounds);
}
pub fn push_arc(&mut self,
transform: &Transform2F,
start_angle: f32,
end_angle: f32,
direction: ArcDirection) {
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if end_angle - start_angle >= PI * 2.0 {
self.push_ellipse(transform);
} else {
let start = vec2f(start_angle.cos(), start_angle.sin());
let end = vec2f(end_angle.cos(), end_angle.sin());
self.push_arc_from_unit_chord(transform, LineSegment2F::new(start, end), direction);
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}
}
pub fn push_arc_from_unit_chord(&mut self,
transform: &Transform2F,
mut chord: LineSegment2F,
direction: ArcDirection) {
let mut direction_transform = Transform2F::default();
if direction == ArcDirection::CCW {
chord *= vec2f(1.0, -1.0);
direction_transform = Transform2F::from_scale(vec2f(1.0, -1.0));
}
let (mut vector, end_vector) = (UnitVector(chord.from()), UnitVector(chord.to()));
for segment_index in 0..4 {
debug!("push_arc_from_unit_chord(): loop segment index {}", segment_index);
let mut sweep_vector = end_vector.rev_rotate_by(vector);
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let last = sweep_vector.0.x() >= -EPSILON && sweep_vector.0.y() >= -EPSILON;
debug!("... end_vector={:?} vector={:?} sweep_vector={:?} last={:?}",
end_vector,
vector,
sweep_vector,
last);
let mut segment;
if !last {
sweep_vector = UnitVector(vec2f(0.0, 1.0));
segment = Segment::quarter_circle_arc();
} else {
segment = Segment::arc_from_cos(sweep_vector.0.x());
}
let half_sweep_vector = sweep_vector.halve_angle();
let rotation = Transform2F::from_rotation_vector(half_sweep_vector.rotate_by(vector));
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segment = segment.transform(&(*transform * direction_transform * rotation));
let mut push_segment_flags = PushSegmentFlags::UPDATE_BOUNDS;
if segment_index == 0 {
push_segment_flags.insert(PushSegmentFlags::INCLUDE_FROM_POINT);
}
self.push_segment(&segment, push_segment_flags);
if last {
break;
}
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vector = vector.rotate_by(sweep_vector);
}
const EPSILON: f32 = 0.001;
}
pub fn push_ellipse(&mut self, transform: &Transform2F) {
let segment = Segment::quarter_circle_arc();
let mut rotation;
self.push_segment(&segment.transform(transform),
PushSegmentFlags::UPDATE_BOUNDS | PushSegmentFlags::INCLUDE_FROM_POINT);
rotation = Transform2F::from_rotation_vector(UnitVector(vec2f( 0.0, 1.0)));
self.push_segment(&segment.transform(&(*transform * rotation)),
PushSegmentFlags::UPDATE_BOUNDS);
rotation = Transform2F::from_rotation_vector(UnitVector(vec2f(-1.0, 0.0)));
self.push_segment(&segment.transform(&(*transform * rotation)),
PushSegmentFlags::UPDATE_BOUNDS);
rotation = Transform2F::from_rotation_vector(UnitVector(vec2f( 0.0, -1.0)));
self.push_segment(&segment.transform(&(*transform * rotation)),
PushSegmentFlags::UPDATE_BOUNDS);
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}
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#[inline]
pub fn segment_after(&self, point_index: u32) -> Segment {
debug_assert!(self.point_is_endpoint(point_index));
let mut segment = Segment::none();
segment.baseline.set_from(self.position_of(point_index));
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let point1_index = self.add_to_point_index(point_index, 1);
if self.point_is_endpoint(point1_index) {
segment.baseline.set_to(self.position_of(point1_index));
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segment.kind = SegmentKind::Line;
} else {
segment.ctrl.set_from(self.position_of(point1_index));
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let point2_index = self.add_to_point_index(point_index, 2);
if self.point_is_endpoint(point2_index) {
segment.baseline.set_to(self.position_of(point2_index));
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segment.kind = SegmentKind::Quadratic;
} else {
segment.ctrl.set_to(self.position_of(point2_index));
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segment.kind = SegmentKind::Cubic;
let point3_index = self.add_to_point_index(point_index, 3);
segment.baseline.set_to(self.position_of(point3_index));
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}
}
segment
}
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#[inline]
pub fn hull_segment_after(&self, prev_point_index: u32) -> LineSegment2F {
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let next_point_index = self.next_point_index_of(prev_point_index);
LineSegment2F::new(
self.points[prev_point_index as usize],
self.points[next_point_index as usize],
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)
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}
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#[inline]
pub fn point_is_endpoint(&self, point_index: u32) -> bool {
!self.flags[point_index as usize]
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.intersects(PointFlags::CONTROL_POINT_0 | PointFlags::CONTROL_POINT_1)
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}
#[inline]
pub fn add_to_point_index(&self, point_index: u32, addend: u32) -> u32 {
let (index, limit) = (point_index + addend, self.len());
if index >= limit {
index - limit
} else {
index
}
}
#[inline]
pub fn point_is_logically_above(&self, a: u32, b: u32) -> bool {
let (a_y, b_y) = (self.points[a as usize].y(), self.points[b as usize].y());
a_y < b_y || (a_y == b_y && a < b)
}
#[inline]
pub fn prev_endpoint_index_of(&self, mut point_index: u32) -> u32 {
loop {
point_index = self.prev_point_index_of(point_index);
if self.point_is_endpoint(point_index) {
return point_index;
}
}
}
#[inline]
pub fn next_endpoint_index_of(&self, mut point_index: u32) -> u32 {
loop {
point_index = self.next_point_index_of(point_index);
if self.point_is_endpoint(point_index) {
return point_index;
}
}
}
#[inline]
pub fn prev_point_index_of(&self, point_index: u32) -> u32 {
if point_index == 0 {
self.len() - 1
} else {
point_index - 1
}
}
#[inline]
pub fn next_point_index_of(&self, point_index: u32) -> u32 {
if point_index == self.len() - 1 {
0
} else {
point_index + 1
}
}
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pub fn transform(&mut self, transform: &Transform2F) {
if transform.is_identity() {
return;
}
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for (point_index, point) in self.points.iter_mut().enumerate() {
*point = *transform * *point;
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union_rect(&mut self.bounds, *point, point_index == 0);
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}
}
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pub fn apply_perspective(&mut self, perspective: &Perspective) {
for (point_index, point) in self.points.iter_mut().enumerate() {
*point = *perspective * *point;
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union_rect(&mut self.bounds, *point, point_index == 0);
}
}
pub fn dilate(&mut self, amount: Vector2F, orientation: Orientation) {
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ContourDilator::new(self, amount, orientation).dilate();
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self.bounds = self.bounds.dilate(amount);
}
fn prepare_for_tiling(&mut self, view_box: RectF) {
// Snap points to the view box bounds. This mops up floating point error from the clipping
// process.
let (mut last_endpoint_index, mut contour_is_monotonic) = (None, true);
for point_index in 0..(self.points.len() as u32) {
if contour_is_monotonic {
if self.point_is_endpoint(point_index) {
if let Some(last_endpoint_index) = last_endpoint_index {
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if !self.curve_with_endpoints_is_monotonic(last_endpoint_index,
point_index) {
contour_is_monotonic = false;
}
}
last_endpoint_index = Some(point_index);
}
}
}
// Convert to monotonic, if necessary.
if !contour_is_monotonic {
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self.make_monotonic();
}
// Update bounds.
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self.bounds = self
.bounds
.intersection(view_box)
.unwrap_or_else(|| RectF::default());
}
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fn make_monotonic(&mut self) {
debug!("--- make_monotonic() ---");
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let contour = self.take();
self.bounds = contour.bounds;
let mut last_endpoint_index = None;
let input_point_count = contour.points.len() as u32;
for point_index in 0..(input_point_count + 1) {
if point_index < input_point_count && !contour.point_is_endpoint(point_index) {
continue;
}
if let Some(last_endpoint_index) = last_endpoint_index {
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let position_index = if point_index == input_point_count {
0
} else {
point_index
};
let baseline = LineSegment2F::new(
contour.points[last_endpoint_index as usize],
contour.points[position_index as usize],
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);
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let point_count = point_index - last_endpoint_index + 1;
if point_count == 3 {
let ctrl_point_index = last_endpoint_index as usize + 1;
let ctrl_position = &contour.points[ctrl_point_index];
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handle_cubic(
self,
&Segment::quadratic(baseline, *ctrl_position).to_cubic(),
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);
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} else if point_count == 4 {
let first_ctrl_point_index = last_endpoint_index as usize + 1;
let ctrl_position_0 = &contour.points[first_ctrl_point_index + 0];
let ctrl_position_1 = &contour.points[first_ctrl_point_index + 1];
let ctrl = LineSegment2F::new(*ctrl_position_0, *ctrl_position_1);
handle_cubic(self, &Segment::cubic(baseline, ctrl));
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}
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self.push_point(
contour.points[position_index as usize],
PointFlags::empty(),
false,
);
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}
last_endpoint_index = Some(point_index);
}
fn handle_cubic(contour: &mut Contour, segment: &Segment) {
debug!("handle_cubic({:?})", segment);
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match segment.as_cubic_segment().y_extrema() {
(Some(t0), Some(t1)) => {
let (segments_01, segment_2) = segment.as_cubic_segment().split(t1);
let (segment_0, segment_1) = segments_01.as_cubic_segment().split(t0 / t1);
contour.push_segment(&segment_0, PushSegmentFlags::empty());
contour.push_segment(&segment_1, PushSegmentFlags::empty());
contour.push_segment(&segment_2, PushSegmentFlags::empty());
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}
(Some(t0), None) | (None, Some(t0)) => {
let (segment_0, segment_1) = segment.as_cubic_segment().split(t0);
contour.push_segment(&segment_0, PushSegmentFlags::empty());
contour.push_segment(&segment_1, PushSegmentFlags::empty());
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}
(None, None) => contour.push_segment(segment, PushSegmentFlags::empty()),
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}
}
}
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fn curve_with_endpoints_is_monotonic(
&self,
start_endpoint_index: u32,
end_endpoint_index: u32,
) -> bool {
let start_position = self.points[start_endpoint_index as usize];
let end_position = self.points[end_endpoint_index as usize];
if start_position.x() <= end_position.x() {
for point_index in start_endpoint_index..end_endpoint_index {
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if self.points[point_index as usize].x() > self.points[point_index as usize + 1].x()
{
return false;
}
}
} else {
for point_index in start_endpoint_index..end_endpoint_index {
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if self.points[point_index as usize].x() < self.points[point_index as usize + 1].x()
{
return false;
}
}
}
if start_position.y() <= end_position.y() {
for point_index in start_endpoint_index..end_endpoint_index {
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if self.points[point_index as usize].y() > self.points[point_index as usize + 1].y()
{
return false;
}
}
} else {
for point_index in start_endpoint_index..end_endpoint_index {
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if self.points[point_index as usize].y() < self.points[point_index as usize + 1].y()
{
return false;
}
}
}
true
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}
// Use this function to keep bounds up to date when mutating paths. See `Outline::transform()`
// for an example of use.
pub(crate) fn update_bounds(&self, bounds: &mut Option<RectF>) {
*bounds = Some(match *bounds {
None => self.bounds,
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Some(bounds) => bounds.union_rect(self.bounds),
})
}
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}
impl Debug for Contour {
fn fmt(&self, formatter: &mut Formatter) -> fmt::Result {
for (segment_index, segment) in self.iter(ContourIterFlags::IGNORE_CLOSE_SEGMENT)
.enumerate() {
if segment_index == 0 {
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write!(
formatter,
"M {} {}",
segment.baseline.from_x(),
segment.baseline.from_y()
)?;
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}
match segment.kind {
SegmentKind::None => {}
SegmentKind::Line => {
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write!(
formatter,
" L {} {}",
segment.baseline.to_x(),
segment.baseline.to_y()
)?;
}
SegmentKind::Quadratic => {
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write!(
formatter,
" Q {} {} {} {}",
segment.ctrl.from_x(),
segment.ctrl.from_y(),
segment.baseline.to_x(),
segment.baseline.to_y()
)?;
}
SegmentKind::Cubic => {
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write!(
formatter,
" C {} {} {} {} {} {}",
segment.ctrl.from_x(),
segment.ctrl.from_y(),
segment.ctrl.to_x(),
segment.ctrl.to_y(),
segment.baseline.to_x(),
segment.baseline.to_y()
)?;
}
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}
}
if self.closed {
write!(formatter, " z")?;
}
Ok(())
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}
}
#[derive(Clone, Copy, Debug, Default, PartialEq, Eq, PartialOrd, Ord)]
pub struct PointIndex(u32);
impl PointIndex {
#[inline]
pub fn new(contour: u32, point: u32) -> PointIndex {
debug_assert!(contour <= 0xfff);
debug_assert!(point <= 0x000f_ffff);
PointIndex((contour << 20) | point)
}
#[inline]
pub fn contour(self) -> u32 {
self.0 >> 20
}
#[inline]
pub fn point(self) -> u32 {
self.0 & 0x000f_ffff
}
}
pub struct ContourIter<'a> {
contour: &'a Contour,
index: u32,
flags: ContourIterFlags,
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}
impl<'a> Iterator for ContourIter<'a> {
type Item = Segment;
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#[inline]
fn next(&mut self) -> Option<Segment> {
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let contour = self.contour;
let include_close_segment = self.contour.closed &&
!self.flags.contains(ContourIterFlags::IGNORE_CLOSE_SEGMENT);
if (self.index == contour.len() && !include_close_segment) ||
self.index == contour.len() + 1 {
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return None;
}
let point0_index = self.index - 1;
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let point0 = contour.position_of(point0_index);
if self.index == contour.len() {
let point1 = contour.position_of(0);
self.index += 1;
return Some(Segment::line(LineSegment2F::new(point0, point1)));
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}
let point1_index = self.index;
self.index += 1;
let point1 = contour.position_of(point1_index);
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if contour.point_is_endpoint(point1_index) {
return Some(Segment::line(LineSegment2F::new(point0, point1)));
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}
let point2_index = self.index;
let point2 = contour.position_of(point2_index);
self.index += 1;
if contour.point_is_endpoint(point2_index) {
return Some(Segment::quadratic(LineSegment2F::new(point0, point2), point1));
}
let point3_index = self.index;
let point3 = contour.position_of(point3_index);
self.index += 1;
debug_assert!(contour.point_is_endpoint(point3_index));
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return Some(Segment::cubic(
LineSegment2F::new(point0, point3),
LineSegment2F::new(point1, point2),
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));
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}
}
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#[derive(Clone, Copy, Debug, PartialEq)]
pub enum ArcDirection {
CW,
CCW,
}
bitflags! {
pub struct ContourIterFlags: u8 {
const IGNORE_CLOSE_SEGMENT = 1;
}
}
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#[inline]
pub(crate) fn union_rect(bounds: &mut RectF, new_point: Vector2F, first: bool) {
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if first {
*bounds = RectF::from_points(new_point, new_point);
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} else {
*bounds = bounds.union_point(new_point)
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}
}